Empty setIn mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set. Any set other than the empty set is called non-empty. In some textbooks and popularizations, the empty set is referred to as the "null set".
Finite setIn mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set.
Quasi-categoryIn mathematics, more specifically , a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a . The study of such generalizations is known as . Quasi-categories were introduced by . André Joyal has much advanced the study of quasi-categories showing that most of the usual basic and some of the advanced notions and theorems have their analogues for quasi-categories.
Monad (category theory)In , a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a in the of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories.
Category of small categoriesIn mathematics, specifically in , the category of small categories, denoted by Cat, is the whose objects are all and whose morphisms are functors between categories. Cat may actually be regarded as a with natural transformations serving as 2-morphisms. The initial object of Cat is the empty category 0, which is the category of no objects and no morphisms. The terminal object is the terminal category or trivial category 1 with a single object and morphism. The category Cat is itself a , and therefore not an object of itself.
Simplicial setIn mathematics, a simplicial set is an object composed of simplices in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and . Formally, a simplicial set may be defined as a contravariant functor from the to the . Simplicial sets were introduced in 1950 by Samuel Eilenberg and Joseph A. Zilber. Every simplicial set gives rise to a "nice" topological space, known as its geometric realization.
Adjoint functorsIn mathematics, specifically , adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e.
Group actionIn mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it.
FunctorIn mathematics, specifically , a functor is a mapping between . Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which is applied.
Category theoryCategory theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in almost all areas of mathematics. In particular, numerous constructions of new mathematical objects from previous ones that appear similarly in several contexts are conveniently expressed and unified in terms of categories.
Automorphism groupIn mathematics, the automorphism group of an object X is the group consisting of automorphisms of X under composition of morphisms. For example, if X is a finite-dimensional vector space, then the automorphism group of X is the group of invertible linear transformations from X to itself (the general linear group of X). If instead X is a group, then its automorphism group is the group consisting of all group automorphisms of X. Especially in geometric contexts, an automorphism group is also called a symmetry group.
Abelian groupIn mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.
Dihedral groupIn mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry. The notation for the dihedral group differs in geometry and abstract algebra. In geometry, D_n or Dih_n refers to the symmetries of the n-gon, a group of order 2n. In abstract algebra, D_2n refers to this same dihedral group.
Abstract simplicial complexIn combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1).
Group theoryIn abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen as groups endowed with additional operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right.
AlgebraAlgebra () is the study of variables and the rules for manipulating these variables in formulas; it is a unifying thread of almost all of mathematics. Elementary algebra deals with the manipulation of variables (commonly represented by Roman letters) as if they were numbers and is therefore essential in all applications of mathematics. Abstract algebra is the name given, mostly in education, to the study of algebraic structures such as groups, rings, and fields.
Category (mathematics)In mathematics, a category (sometimes called an abstract category to distinguish it from a ) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the , whose objects are sets and whose arrows are functions. is a branch of mathematics that seeks to generalize all of mathematics in terms of categories, independent of what their objects and arrows represent.
T-structureIn the branch of mathematics called homological algebra, a t-structure is a way to axiomatize the properties of an of a . A t-structure on consists of two subcategories of a or stable which abstract the idea of complexes whose cohomology vanishes in positive, respectively negative, degrees. There can be many distinct t-structures on the same category, and the interplay between these structures has implications for algebra and geometry. The notion of a t-structure arose in the work of Beilinson, Bernstein, Deligne, and Gabber on perverse sheaves.
Category of setsIn the mathematical field of , the category of sets, denoted as Set, is the whose are sets. The arrows or morphisms between sets A and B are the total functions from A to B, and the composition of morphisms is the composition of functions. Many other categories (such as the , with group homomorphisms as arrows) add structure to the objects of the category of sets and/or restrict the arrows to functions of a particular kind.
Nerve (category theory)In , a discipline within mathematics, the nerve N(C) of a C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called the classifying space of the category C. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory. The nerve of a category is often used to construct topological versions of moduli spaces.
